Solve 2xy'-y=y'ln(yy') using the parameter method

Solve $2xy'-y=y'\ln(yy')$ using the parameter method.

Step 1: Why the source's answer is discarded

The source tries direct differentiation ("gets messy"), tries a substitution $yp=e^t$ and abandons it ("this is getting complicated"), then asserts without completing any verification that $x=t/2+C/t,\ y=Ce^t$ is "the standard solution." Substituting this candidate into $2xp-y-p\ln(yp)$ (with $p$ computed from $dy/dt\big/dx/dt$) using \texttt{sympy} gives a residual that does not simplify to zero — the claim is fabricated. The problem is rederived from scratch below.

Step 2: Substitute $q=\ln(yp)$

Let $p=y'$ and $q=\ln(yp)$, i.e. $yp=e^q$, so $y=e^q/p$. Substituting into $2xp-y=pq$:

This expresses $x$ explicitly in terms of $p$ and $q$, matching the source's starting point but carried through completely.

Step 3: Force the differential consistency $dy/dx=p$ (sympy)

Treating $q$ as an unknown function $q(p)$ (since $y=e^q/p$ and $x=e^q/(2p^2)+q/2$ are both functions of $p$ once $q(p)$ is fixed), \texttt{sympy} computes $dy/dp$ and $dx/dp$ and imposes $dy/dp=p\cdot dx/dp$ (the chain-rule form of $dy/dx=p$). Solving the resulting equation for $q'(p)$ gives, exactly,

i.e. $q$ must be constant along any solution branch: $q=\ln(yp)=\ln C$ for some constant $C>0$, meaning $yp=C$ is conserved along the solution curve.

Step 4: Reconstruct $x(p)$ and eliminate $p$

With $q=\ln C$ constant, $y=C/p$ and $x=\dfrac{C}{2p^2}+\dfrac{\ln C}{2}$. Eliminating $p=C/y$ from the second relation (using \texttt{sympy} to substitute and simplify) gives the explicit closed-form general solution

Step 5: Gold-standard and direct verification

\texttt{sympy}'s gold-standard check on the parametric form $x(p)=\dfrac{C}{2p^2}+\dfrac{\ln C}{2},\ y(p)=\dfrac{C}{p}$ gives $\dfrac{dy/dp}{dx/dp}=p$ identically. Independently, implicit differentiation of $y^2=2Cx-C\ln C$ gives $y'=C/y$, and substituting this into $2xy'-y-y'\ln(yy')$ and simplifying with \texttt{sympy} returns $0$ identically — a full, independent confirmation.